Problem 12 - College Algebra

Question

In Exercises \(9-16,\) find: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rrr} 3 & 1 & 1 \\ -1 & 2 & 5 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & -3 & 6 \\ -3 & 1 & -4 \end{array}\right] $$

Step-by-Step Solution

Answer

The solutions for \(A + B\), \(A - B\), \(-4A\) and \(3A + 2B\) are \[ \begin{bmatrix} 5 & -2 & 7 \ -4 & 3 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 4 & -5 \ 2 & 1 & 9\end{bmatrix}, \begin{bmatrix} -12 & -4 & -4 \ 4 & -8 & -20\end{bmatrix}, \begin{bmatrix} 13 & -3 & 15\-9 & 8 & 23\end{bmatrix} \]respectively.

Step 1: Calculating \(A + B\)

For performing the operation \(A + B\) we simply add the elements in the same positions from both matrices. Result: \[ \begin{bmatrix} 3+2 & 1+(-3) & 1+6\ -1+(-3) & 2+1 & 5+(-4) \end{bmatrix} = \begin{bmatrix} 5 & -2 & 7 \ -4 & 3 & 1 \end{bmatrix} \]

Step 2: Calculating \(A - B\)

For performing the operation \(A - B\) we subtract the elements in the same positions in B from A. Result: \[ \begin{bmatrix} 3-2 & 1-(-3) & 1-6\ -1-(-3) & 2-1 & 5-(-4) \end{bmatrix} = \begin{bmatrix} 1 & 4 & -5 \ 2 & 1 & 9 \end{bmatrix} \]

Step 3: Calculating \(-4A\)

When performing scalar multiplication of matrix A and the scalar -4, every element of the matrix A is multiplied by the scalar. Result: \[ -4*\begin{bmatrix} 3 & 1 & 1 \ -1 & 2 & 5 \end{bmatrix} = \begin{bmatrix} -12 & -4 & -4 \ 4 & -8 & -20 \end{bmatrix} \]

Step 4: Calculating \(3A + 2B\)

The operation \(3A + 2B\) is carried out as scalar multiplication followed by matrix addition. First, A is multiplied by 3 and B is multiplied by 2. The resulting matrices are then added component-wise. Result is \[ 3*\begin{bmatrix} 3 & 1 & 1 \ -1 & 2 & 5 \end{bmatrix}+ 2*\begin{bmatrix} 2 & -3 & 6\ -3 & 1 & -4 \end{bmatrix}= \begin{bmatrix} 9+4 & 3-6 & 3+12\ -3-6 & 6+2 & 15+8 \end{bmatrix}= \begin{bmatrix} 13 & -3 & 15 \ -9 & 8 & 23 \end{bmatrix} \]

Key Concepts

Matrix Addition

Matrix addition involves the addition of two matrices by adding their corresponding elements.
The matrices must have the same dimensions.
Each element in the first matrix is added to the element in the second matrix that is in the same position. A simple way to visualize this is to imagine stacking both matrices on top of one another, aligning them perfectly.
Then, just add the numbers that sit on top of each other:

• If you have: \( A = \begin{bmatrix} 3 & 1 & 1 \ -1 & 2 & 5 \end{bmatrix} \)
• And \( B = \begin{bmatrix} 2 & -3 & 6 \ -3 & 1 & -4 \end{bmatrix} \)
• The sum \( A + B \) would be: \( \begin{bmatrix} 3+2 & 1+(-3) & 1+6 \ -1+(-3) & 2+1 & 5+(-4) \end{bmatrix} \)
• Resulting in: \( \begin{bmatrix} 5 & -2 & 7 \ -4 & 3 & 1 \end{bmatrix} \)

This simple operation is a foundational concept in linear algebra, helping set the stage for more complex matrix manipulations.

Matrix Subtraction

Matrix subtraction is similar to matrix addition and involves subtracting elements in the same positions from two matrices.
Again, the matrices should have identical dimensions.The process follows a straightforward path: just like adding matrices, imagine one matrix lying over the other,
then subtract the bottom element from the top element.

• For example, take the same matrices \( A \) and \( B \) as before.
• When you calculate \( A - B \): \( \begin{bmatrix} 3-2 & 1-(-3) & 1-6 \ -1-(-3) & 2-1 & 5-(-4) \end{bmatrix} \)
• The result will be: \( \begin{bmatrix} 1 & 4 & -5 \ 2 & 1 & 9 \end{bmatrix} \)

Subtraction helps refine our understanding of the spatial orientation of matrices and sets the stage for more advanced operations and transformations.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a single number, known as a scalar.
This operation changes the magnitude of the matrix elements without altering the matrix's size or structure.To perform scalar multiplication:

• Consider a scalar \( k \) and a matrix \( A \).
• Multiply each element of matrix \( A \) by \( k \):
• For example, if \( k = -4 \) and \( A = \begin{bmatrix} 3 & 1 & 1 \ -1 & 2 & 5 \end{bmatrix} \):
• The result is \(-4 * \begin{bmatrix} 3 & 1 & 1 \ -1 & 2 & 5 \end{bmatrix} = \begin{bmatrix} -12 & -4 & -4 \ 4 & -8 & -20 \end{bmatrix} \)

Scalar multiplication is crucial when scaling transformations or modulating the effect a matrix induces in linear transformations.

Linear Algebra Concepts

Linear algebra is the mathematical study of vectors, vector spaces, linear transformations, and matrices.
Matrices serve as tools to handle linear equations and transformations and are essential for solving problems in engineering, physics, and computer science. Key operations in linear algebra include:

• Matrix Addition and Subtraction: Useful for combining or differentiating between linear equations.
• Scalar Multiplication: Alters the scale of a matrix's impact on transformation processes.

These foundational operations support more advanced concepts like eigenvalues, eigenvectors, and matrix decompositions.
Understanding these basics opens up a world of computational efficiency and problem-solving capabilities in diverse practical fields.